scholarly journals Two-Stage Adaptive Optimal Design with Fixed First-Stage Sample Size

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Adam Lane ◽  
Nancy Flournoy
Keyword(s):  
Maximum Likelihood ◽  
Optimal Design ◽  
Sample Size ◽  
Asymptotic Distribution ◽  
Fixed Number ◽  
Maximum Likelihood Estimates ◽  
Small Pilot Study ◽  
Distribution Of The Maximum ◽  
Optimal Procedures ◽  
Mean Function

In adaptive optimal procedures, the design at each stage is an estimate of the optimal design based on all previous data. Asymptotics for regular models with fixed number of stages are straightforward if one assumes the sample size of each stage goes to infinity with the overall sample size. However, it is not uncommon for a small pilot study of fixed size to be followed by a much larger experiment. We study the large sample behavior of such studies. For simplicity, we assume a nonlinear regression model with normal errors. We show that the distribution of the maximum likelihood estimates converges to a scale mixture family of normal random variables. Then, for a one parameter exponential mean function we derive the asymptotic distribution of the maximum likelihood estimate explicitly and present a simulation to compare the characteristics of this asymptotic distribution with some commonly used alternatives.

scholarly journals Asymptotic Distribution of the Maximum Likelihood Estimator for a Stochastic Frontier Function Model with a Singular Information Matrix

1993 ◽  
Vol 9 (3) ◽  
pp. 413-430 ◽  
Author(s):  
Lung-Fei Lee
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimator ◽  
Asymptotic Distribution ◽  
Information Matrix ◽  
Stochastic Frontier ◽  
Maximum Likelihood Estimates ◽  
Ratio Test ◽  
Likelihood Estimator ◽  
True Parameter ◽  
Distribution Of The Maximum

This paper investigates the asymptotic distribution of the maximum likelihood estimator in a stochastic frontier function when the firms are all technically efficient. For such a situation the true parameter vector is on the boundary of the parameter space, and the scores are linearly dependent. The asymptotic distribution of the maximum likelihood estimator is shown to be a mixture of certain truncated distributions. The maximum likelihood estimates for different parameters may have different rates of stochastic convergence. The model can be reparameterized into one with a regular likelihood function. The likelihood ratio test statistic has the usual mixture of chi-square distributions as in the regular case.

scholarly journals Sample Size Requirements for Zero-Order Models

1980 ◽  
Vol 17 (2) ◽  
pp. 221-227 ◽  
Author(s):  
Manohar U. Kalwani ◽  
Donald G. Morrison
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Sample Size ◽  
Market Share ◽  
Zero Order ◽  
Maximum Likelihood Estimates ◽  
Model Estimation ◽  
Order Model ◽  
Model Parameter ◽  
Polarization Index

The authors report the sample sizes required to obtain maximum likelihood estimates of parameters in a zero-order model. In the typical formulation of the model, parameter estimation requires about 2000 individuals with five purchases per consumer. The original parameters can be easily transformed, however, into a market share estimate and a polarization index that convey the key information sought from the model. Estimation of the transformed parameters requires smaller samples of about 400 individuals with five purchases per consumer.

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